As test statistics we consider \(\rho _H(\hat{\varphi }_{I,n},\widetilde{\varphi }_{I,0})^2\)  and \(\rho _H(\hat{\varphi }_{G,n},\widetilde{\varphi }_{G,0})^2\) , where \(\widetilde{\varphi }_{I,0}\)  and \(\widetilde{\varphi }_{G,0}\)  denote the best approximations with respect to the Hellinger distance of \(\varphi _{I,0}\)  and \(\varphi _{G,0}\)  through Laguerre polynomials, respectively. In order to choose the degrees of the approximation we use in the same way as before the BIC. The resulting values are shown in Table REF  and it can be seen that \(m_1=2\)  and \(m_2=1\)  yields the smallest BIC. While Figure REF  shows that \(m_1=2\)  yields a reasonable approximation to \(\varphi _{I,0}\) , we see that \(m_2=1\)  is a bit too small to represent \(\varphi _{G,0}\)  well. Therefore, it is very important that we compare the estimate with the closest Laguerre-type density \(\widetilde{\varphi }_{G,0}\)  rather than with \(\varphi _{G,0}\)  directly. The fact that the BIC suggests such a small model indicates that \(n=40\)  is too small to properly differentiate the form of \(\varphi _{G,0}\)  suggested by [1]} from other models. Therefore, the power of our test would be better if we had more observations.
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